On Weighted Compactness of Commutators of Bilinear Vector-valued Singular Integral Operators and Applications

Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderon-Zygmund classes. For any b(->)=(b(1), b(2))is an element of(CMO(R-n))(2), let [T, b(j)](ej) (j=1, 2), [T, b(->)](alpha) be the commutators in the j-th entry and the iterated commutators of T, respectively. In this paper, for all p(0) > 1, p0/2<p)](alpha) are weighted compact operators from L-p1(w(1))xL(p2)(w(2)) to L-p(nu w(->)), where w(->)=(w(1), w(2))is an element of A(p)(->)/p0 and nu w(->)=w(1)(p/p1)w(2)(p/p2). As applications, we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood-Paley square operators with some mild kernel regularity, including bilinear g function, bilinear g(lambda)* function and bilinear Lusin's area integral. In addition, we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear g function, bilinear g(lambda)* function and bilinear Lusin's area integral, respectively.